Approaches to the design of model predictive controllers for linear, piecewise linear and nonlinear systems

APPROACHES TO THE DESIGN OF MODEL PREDICTIVE CONTROLLERS FOR LINEAR, PIECEWISE LINEAR AND NONLINEAR SYSTEMS SUI DAN (B.Eng., M. Eng., Northwest Polytechnical University) A DISSERTATION SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2006 i Acknowledgments I would like to express my sincere appreciation to my supervisor, Assoc. Prof. Ong Chong Jin, for his invaluable guidance, insightful comments, strong encouragements and personal concerns both academically and otherwise throughout the course of the research. I benefit a lot from his comments and critiques. I would also like to thank Dr. S. Sathiya Keerthi and Prof. Elmer G. Gilbert, who have given me invaluable suggestions for this research. I gratefully acknowledge the financial support provided by the National University of Singapore through Research Scholarship that makes it possible for me to study for academic purpose. Thanks are also given to my friends and technicians in Mechatronics and Control Lab for their support and encouragement. They have provided me with helpful comments, great friendship and a warm community during the past few years in NUS. Finally, my deepest thanks go to my parents, for their encouragements, moral supports and loves. Special thanks to Feng Le for our happy time. NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE ii Table of Contents Acknowledgments i Summary vii List of Tables ix List of Figures xi List of Symbols xii Acronyms xiv Introduction 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 MPC for Linear Systems with Bounded Disturbances . . . . . . 1.2.2 MPC for Piecewise Linear/Affine Systems with Bounded Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear MPC of Low Computational Complexity . . . . . . . 1.3 Objectives and Scope of the Thesis . . . . . . . . . . . . . . . . . . . . 1.4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE TABLE OF CONTENTS 1.5 iii 1.4.1 Multi-mode MPC Controller for Constrained LBD Systems . . 1.4.2 Controller Design for Constrained PWLBD Systems . . . . . . 1.4.3 Computations of Disturbance Invariant Sets for PWLBD Systems 1.4.4 Nonlinear MPC via Support Vector Machine . . . . . . . . . . Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . Definitions, Set Operations and Procedures 11 2.1 Polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Operations on Polytope . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Inner Polytopal Approximation . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Multi-parametric Programming . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Invariant Sets of Constrained Linear Systems . . . . . . . . . . . . . . 23 Multi-mode MPC Controller for Constrained LBD Systems 26 3.1 Single-mode Robust MPC Controller . . . . . . . . . . . . . . . . . . . 27 3.2 Approximation of F∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Multi-mode Robust MPC Controller . . . . . . . . . . . . . . . . . . . 31 3.3.1 Off-line Computation of State-feedback Controller . . . . . . . 31 3.3.2 Multi-mode MPC Controller Design . . . . . . . . . . . . . . 32 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Computation of d-invariant Sets of Constrained PWLBD Systems 43 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE TABLE OF CONTENTS 4.3 4.4 4.5 iv Properties and Approximation of F∞ . . . . . . . . . . . . . . . . . . . 47 4.3.1 Properties of F∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3.2 Outer Approximation of F∞ . . . . . . . . . . . . . . . . . . . 50 4.3.3 Reachable Set Operation . . . . . . . . . . . . . . . . . . . . . 54 Computation of Constraint Admissible d-invariant Sets . . . . . . . . . 56 4.4.1 Maximal d-invariant Set . . . . . . . . . . . . . . . . . . . . . 56 4.4.2 Computation of Constraint Admissible, Polytopal d-invariant Sets 57 4.4.3 Enlargement of Constraint Admissible, Polytopal d-invariant Sets 59 4.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Time Sub-optimal Control for Constrained PWLBD Systems 65 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.3 Derivation of Nominal Controller . . . . . . . . . . . . . . . . . . . . . 67 5.3.1 Design via Lyapunov Methods . . . . . . . . . . . . . . . . . . 67 5.3.2 Design via Singular Value Method . . . . . . . . . . . . . . . 70 5.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Robust Time Optimal Control . . . . . . . . . . . . . . . . . . . . . . 74 5.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 74 5.4.2 Time Sub-optimal Controller Design . . . . . . . . . . . . . . 75 5.4.3 Robust Closed-loop Stability . . . . . . . . . . . . . . . . . . . 77 5.4.4 State Feedback Solution to Proposed Controller . . . . . . . . . 78 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.4 5.5 NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE TABLE OF CONTENTS 5.6 v Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Predictive Control for Constrained PWLBD Systems 85 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.2 Robust Model Predictive Control . . . . . . . . . . . . . . . . . . . . . 86 6.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 86 6.2.2 Robust Closed-loop Stability . . . . . . . . . . . . . . . . . . . 90 6.2.3 State Feedback Solution to Proposed Controller . . . . . . . . . 91 6.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Model Predictive Control for Nonlinear Systems via Support Vector Machine 97 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2 Stability of Nonlinear MPC . . . . . . . . . . . . . . . . . . . . . . . . 99 7.3 Characterization of Terminal Set . . . . . . . . . . . . . . . . . . . . . 102 7.4 84 7.3.1 Choice of Terminal Set . . . . . . . . . . . . . . . . . . . . . . 102 7.3.2 SVC for Characterizing X f . . . . . . . . . . . . . . . . . . . . 103 Characterization of Terminal Cost . . . . . . . . . . . . . . . . . . . . 107 7.4.1 Choice of Terminal Cost . . . . . . . . . . . . . . . . . . . . . 107 7.4.2 SVR for Characterizing F . . . . . . . . . . . . . . . . . . . . 108 7.5 Feasibility Enforcement . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Conclusion NATIONAL UNIVERSITY OF SINGAPORE 121 SINGAPORE TABLE OF CONTENTS vi 8.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.2 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 123 Bibliography NATIONAL UNIVERSITY OF SINGAPORE 125 SINGAPORE vii Summary The thesis is concerned with improving the performance and robustness of model predictive control (MPC) controllers for (1) constrained linear systems with bounded disturbances (LBD systems); (2) constrained piecewise linear systems with bounded disturbances (PWLBD systems); (3) constrained nonlinear systems. A multi-mode MPC controller is proposed for constrained LBD systems that guarantees constraint satisfaction and robust closed-loop stability. The design achieves the objective of having a large domain of attraction, good asymptotic behavior and reasonably low online computation. Furthermore, the proposed controller can be determined off-line. For constrained PWLBD systems, two approaches are proposed under the time optimal control (TOC) and MPC frameworks. Both approaches result in the polytopal domains of attraction using an inner polytopal approximation. The resulting control laws of these two approaches can guarantee robust closed-loop stability and can also be determined off-line, which in sequence leads to reasonable on-line computational requirement. Disturbance invariant sets play an important role for the controller design of constrained PWLBD systems. One of the contributions of this thesis is the development of several algorithms for computing disturbance invariant sets and their approximations for PWLBD systems. For constrained nonlinear systems, an approach is proposed to approximate the terminal set and the terminal cost off-line using support vector machine (SVM). SVM is a powerful pattern recognition technique and the approach exploits the flexibility in the choices of the terminal set and cost and is less demanding in terms of the approximat- NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE SUMMARY viii ing accuracy. The resulting terminal set is large and, hence provides a large domain of attraction. NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE ix List of Tables 3.1 Results for selected values of k for these two examples. . . . . . . . . . 31 4.1 Results for selected values of k. 54 7.1 Comparison of the shortest possible horizon (N). . . . . . . . . . . . . 118 7.2 The shortest possible horizon (N), optimal performance index (J) and . . . . . . . . . . . . . . . . . . . . . the CPU time (t) over 100 time steps of the proposed controller. . . . . 119 NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 7.6 Examples 118 No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Initial states x(0) [-0.683, -0.864] [-0.523, 0.744] [0.808, 0.121] [0.774, -0.222] [0.292, 0.228] [-0.08, -0.804] [1.000,-0.500] [3.800,-3.000] [4.000,-1.000] [1.000,-3.000] NA 1 NB 14 13 10 10 12 Table 7.1: Comparison of the shortest possible horizon (N). of the pendulum ℓˆ and gravitational acceleration g. ˆ The values of the parameters are ˆ ˆ m= ˆ M=0.5, ℓ=1.4, g=10 ˆ and the constraints are: U = {u ∈ R : |u| ≤ 2}, X = {x ∈ R4 : |x1 | ≤ 6, |x2 | ≤ 5, |x3 | ≤ π /4, |x4 | ≤ 5}. The stage cost ℓ(x, u)=xT Qx+uT Ru is used with Q=diag(1, 2, 1, 0.5), R=3.2 and the gain of stabilizing linear feedback controller K=[0.559, 1.647, 28.67, 8.664]. The sampling period is 0.1s. The characterization of X f uses 1930 example points adaptively selected from a set of 9786 points. The total time needed for the overall tuning/training exercise is about ˜ hours and O(x) is determined by 345 support vectors with b = 11.82 and γ˜ =0.12. The optimal Gaussian kernel width σˆ = 0.9. FR is obtained using 819 example points selected from a total of 4122 and the total time needed for the overall tuning/training ˆ exercise is about 4.8 hours. For these training data, N=39. Table 7.2 and Figure 7.5 show the performance of our approach on this example for selected starting points. For NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 7.6 Examples 119 each starting point, 100 steps (10 seconds) are performed. No. 1. 2. 3. Initial states x(0) [5, -2, 0.2, -0.4] [-6, 2.2, -.5, 2] [-5.3, 2.7, -0.7, 2.5] N J 475.78 1458.32 2110.35 t(s) 45.483 75.063 127.459 Table 7.2: The shortest possible horizon (N), optimal performance index (J) and the CPU time (t) over 100 time steps of the proposed controller. x1 −5.3 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 Time(seconds) 70 80 90 100 x2 2.7 0.5 x3 −0.7 −1 x4 2.5 −1 Figure 7.5: Closed-loop responses of MPC starting from point 3. NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 7.7 Summary 7.7 120 Summary This chapter shows the application of SVM learning to the implementation of MPC for constrained nonlinear systems. The resulting terminal set, learned by support vector classification, is much larger than those seen in the literature. Consequently, the domain of attraction under MPC is greatly enlarged. When the state constrained set is compact, the large terminal set also leads to a lower on-line computational effort via the use of a shorter horizon. In the examples considered, this reduction can be as much as an order of magnitude. The off-line computation grows rapidly with the order of the system, but based on the experience of the examples considered and others, the approach is expected to handle nonlinear system up to order nx = 6. NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 121 Chapter Conclusion This thesis provides several approaches to the design of MPC controllers for constrained LBD, PWLBD and nonlinear systems. Most practical systems, such as engineering, biological and economic systems, are often described by linear models with additive disturbances. Although MPC for linear models is widely adopted in recent years, its expensive on-line computational requirement limits its potential applications. A framework for multi-mode MPC controllers for LBD systems is proposed to reduce the on-line computational cost and simultaneously provide good asymptotical behavior of systems. This thesis also considers one special class of nonlinear systems, PWL systems, as many nonlinear systems can be approximated closely by PWL models. Two approaches are presented for the design of stabilizing control laws that preserve the convexity of domains of attraction and have reasonable on-line computational load. MPC design for general nonlinear systems is also considered. By using a large terminal set derived using SVM, the proposed approach can relax the on-line computational burden using a short N. The main research contributions reported in this thesis are summarized below. NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 8.1 Contributions 8.1 122 Contributions Multi-mode MPC Controller for Constrained LBD Systems A multi-mode MPC controller approach is proposed for constrained LBD systems in Chapter 3. It has the advantage of combining the merits of the underlying single-mode controllers resulting in a system with large domain of attractions and good asymptotic performance while avoiding the associated problem of having many partitions for the feedback control law. The work in this research is of considerable importance since various single controllers can be put together under the proposed multi-mode framework. The domain of attraction is the union of all domains of attraction of the constituent single controllers while the asymptotic behavior is the best among them. Under this multi-mode framework, the low computation is needed. Computations of Disturbance Invariant Sets for Constrained PWLBD Systems In Chapter 2, a procedure to approximate a P-collection in any finite dimension by an inner polytope is presented. This approach helps to compute polytopal d-invariant sets and approximate domains of attraction for constrained PWLBD systems. In Chapter 4, the existence of the minimal d-invariant set F∞ of PWLBD systems is shown and an algorithm is proposed for computing its polytopal d-invariant outer bounds. Since F∞ is generally a P-collection, the polytopal outer bounds of F∞ may not truly represent F∞ . However, its outer bounds are useful for computational purposes. Several algorithms are also presented to compute constraint admissible, polytopal d-invariant sets of PWLBD systems. These sets are used as terminal sets in MPC and TOC approaches for PWLBD systems in Chapters and to lower the computational complexity. Controller Design for Constrained PWLBD Systems In Chapter 5, a relatively simple method for designing stabilizing PWL feedback controllers for nominal PWL system is described. The advantage is that the design considers only one dynamic of PWL system at a time and tries to keep each Ki small so that the NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 8.2 Directions for Future Work 123 control constraints are less likely to be binding, resulting in a large maximal d-invariant set. The disadvantage is that the feasible design space may be small. In Chapters and 6, two approaches for constrained PWLBD systems are proposed. Both constraint admissible, polytopal d-invariant sets and the inner polytopal approximation operation are used in these approaches. Each of the resulting robust control laws has a piecewise affine structure and can be explicitly determined off-line. However, both approaches propose a sub-optimal solution. MPC for Constrained Nonlinear Systems via SVM In Chapter 7, the application of SVM learning to MPC for constrained nonlinear systems is shown. The basic idea is to use SVM learning to find a large terminal set and a corresponding terminal cost. The proposed approach can tolerate inaccuracies in the approximation and has a procedure to ensure the closed-loop stability under reasonable computations. With a large terminal set, the domain of attraction for a fixed horizon is also large. This large domain of attraction means shorter horizon can be used to cover a given region, resulting in a lower computational cost. 8.2 Directions for Future Work Several directions are available for future research based on the work in this thesis. Stochastic Model Predictive Control The systems considered in this thesis deal with bounded disturbances. To design the control laws, the effect of the bounded disturbance is accounted for through the use of strengthened constraint sets of X and U. This results in design conservativeness. Some attempts [24] on design of controllers for linear systems with disturbance described stochastically are currently under way. It is expected that the MPC approaches can handle such a system. NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 8.2 Directions for Future Work 124 Controller Design for Parametric Uncertain Systems The system considered in this research deals with additive disturbances. The possibility of extending the current work to system with parametric uncertainties should be looked into. Robust Nonlinear Model Predictive Control In Chapter 7, we consider the constrained nonlinear systems without disturbances. 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NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE [...]... require the proper handling of the piecewise nature of PWL systems and the effect of disturbances under such a structure One key problem in the controller design for PWLBD systems is the lack of convexity of the domain of attraction These proposed approaches result in the polytopal domains of attraction using an inner polytopal approximation The convex approximation can be used for a union of finite polytopes... conditions for the finite termination of this algorithm are given 1.2.3 Nonlinear MPC of Low Computational Complexity Since most physical systems are highly nonlinear, the performance of MPC based on linear or PWL/PWA models can be poor This has motivated the development of MPC for general nonlinear models with state and input constraints However, the major obstacle for applying MPC to constrained nonlinear systems. .. true for PWLBD systems They are needed in characterizing the asymptotic behaviors of the system and as terminal sets for stability and feasibility of MPC In this thesis, one of the contributions is the development of several algorithms for computing the disturbance invariant sets and their approximations for constrained PWLBD systems 1.4.4 Nonlinear MPC via Support Vector Machine For constrained nonlinear. .. neural networks and the condition of the accuracy of the approximation is given 1.3 Objectives and Scope of the Thesis This thesis attempts to improve and characterize several issues of MPC control law: the domain of attraction, asymptotical behavior and the on-line computational effort These issues are addressed within the scope of the thesis which is restricted to (1) constrained LBD systems; (2) constrained... domain of attraction for a fixed N Increasing the length of N leads to a greater number of decision variables and, therefore, to a greater on-line computational effort One of the ways to reduce the on-line computational effort is to enlarge X f via the use of a shorter horizon For example, in [19], a terminal set is enlarged by using a local LDI representation for a nonlinear system and by solving a linear. .. exploits approximating approaches, such as support vector machine (SVM), for constrained nonlinear systems This chapter provides a review of the literature on such systems 1.1 Background The analysis of physical systems is often done by using mathematical models However, such models are usually idealistic in that they may not capture all the complexities of the real systems and their physical constraints... nonlinear systems, the approximations of the terminal set X f and terminal cost F off-line using SVM are proposed SVM is a pattern recognition technique, both for regression and classification problems The approach exploits the flexibility in the choices of X f and F and is less demanding in terms of the approximating accuracy NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 1.5 Organization of the Thesis 9 The. .. conditions, the proposed approach has fewer partitions of the domain of attraction compared with some standard robust linear MPC approaches [21, 61] NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 1.4 Contributions of the Thesis 8 1.4.2 Controller Design for Constrained PWLBD Systems For constrained PWLBD systems, two approaches are proposed under the time optimal control (TOC) and MPC frameworks Both approaches. .. exactly described by the model [43] Therefore, the issue of linear MPC in the face of uncertainties has received much attention recently Several MPC methods have been proposed for linear systems with bounded disturbances NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 1.2 Literature Review 4 (LBD systems) The simplest [54, 82] is to ignore the disturbances and rely on the inherent robustness of deterministic... large and, hence provides a large domain of attraction Furthermore, a larger terminal set implies faster on-line computational work via the use of a shorter horizon 1.5 Organization of the Thesis This thesis is organized as follows: Chapter 1 introduces the background of MPC and reviews the literature of MPC for constrained LBD, PWLBD and nonlinear systems Chapter 2 reviews some basic concepts and methodologies . APPROACHES TO THE DESIGN OF MODEL PREDICTIVE CONTROLLERS FOR LINEAR, PIECEWISE LINEAR AND NONLINEAR SYSTEMS SUI DAN (B.Eng., M. Eng.,. of the contributions of this thesis is the development of sev- eral algorithms for computing disturbance invariant sets and their approximations for PWLBD systems. For constrained nonlinear systems, . UNIVERSITY OF SINGAPORE SINGAPORE vii Summary The thesis is concerned with improving the performance and robustness of model pre- dictive control (MPC) controllers for (1) constrained linear systems